Struggling with LocalBusiness Schema sameAs Property for Multi-Location JSON-LD Cross-Referencing

We're currently deep into implementing JSON-LD for a SaaS platform that boasts numerous physical locations, each meticulously detailed on its own dedicated landing page. Our overarching goal is quite specific: to establish clear and robust schema hierarchy and relational links, aiming to consolidate review data at the organizational level while simultaneously ensuring that each individual location reaps its full potential in local SEO benefits. This dual objective is proving to be a more intricate challenge than initially anticipated, particularly concerning how Google interprets these complex structured data relationships.

The core technical challenge we're grappling with lies in correctly structuring the LocalBusiness schema for each distinct location and subsequently linking it back to a central Organization schema. Our focus has been intensely on the sameAs property and its intricate interaction with other relational properties like branchOf or parentOrganization. We're consistently encountering ambiguity in how Google's parsers and algorithms interpret internal versus external sameAs links, especially within multi-entity scenarios where clarity in schema hierarchy is paramount for accurate indexing and rich snippet generation.

In our attempts to resolve this, we've explored several implementation strategies. Initially, we tried defining each LocalBusiness entity with a parentOrganization property, pointing directly to the main Organization's @id, assuming this would clearly establish the parent-child relationship. We also experimented with embedding an array of LocalBusiness entities directly within the Organization schema, utilizing either hasPart or subOrganization properties, to see if a containment approach would yield better results. For the crucial sameAs property itself, our tests have included using the canonical URL of the main organization's website, the Google My Business URL associated with the parent entity, and even an internal @id reference pointing to the main Organization entity defined elsewhere on the same page. Furthermore, we've extensively tried employing the branchOf property within the LocalBusiness schema, again pointing to the Organization's @id, believing this would explicitly denote the branch relationship.

Despite these varied and rigorous attempts, and even after passing basic schema validation tools like Google's Rich Results Test without critical errors, we're consistently not observing the desired aggregation of reviews or a clear, explicit indication of location hierarchy within the SERPs. Specifically, our observations suggest that using sameAs to point to the main organization's page doesn't seem to establish the 'branch' or 'parent' relationship as effectively as we had anticipated. More often than not, it appears to be interpreted merely as an external identifier of an equivalent entity, rather than a structural link within our owned, hierarchically organized entities.

Given these persistent issues, we have some specific questions for the community of experienced schema implementers here. First, what is considered the most robust and semantically correct methodology to link individual LocalBusiness schemas to a central Organization schema using @id references and properties such as sameAs, branchOf, or parentOrganization, especially for a complex multi-location setup? Second, regarding the sameAs property on individual LocalBusiness schemas, should it ideally point to the parent Organization's primary URL, its Google My Business profile, or perhaps be entirely omitted if branchOf is already being utilized to define the relationship? Finally, are there any advanced best practices or canonical approaches for ensuring Google truly comprehends the hierarchical relationship we're trying to convey and, crucially, aggregates relevant signals like reviews effectively when dealing with such complex multi-location LocalBusiness structured data?

Any insights, advanced strategies, or real-world examples from seasoned schema implementers would be immensely helpful in navigating this technical block. We are keenly looking for expert guidance on how to avoid these persistent misinterpretations and fully leverage the LocalBusiness schema for our extensive and growing network of locations. Waiting for an expert reply!