How to state equality of 2 identifiers (tokens) pointing at the same resource from 2 existing stacks with dependencies based on Outputs

[KKorzuno]

Hi all. I am working with multiple parsed stacks. There are cross-references based on outputs between them. When I ingest the stacks those outputs are NOT considered for all the different tokenized IDs that the constructs have, meaning even if Resources are added to a RestAPI in a real deployment in AWS, the tokens that I see in CDK seem to have NOTHING in common. Is there a way to enforce that they are somehow considering the outputs? I read a thread from 2 years ago about outputs not being visible after deployment and a resolution with a file-based listing of logical IDs, which I can then use to pull the output from the stack. But I checked there and I do not understand how that works. In the cdk.out I am getting things like:

"/abc-api-dev/abc-api-dev": [{ "type": "aws:cdk:logicalId", "data": "abcapidev" }]

This is not helpful with anything that I have access to in CDK, which looks like this:
This is for RestAPI:

LogicalID: 
${Token[abc-bg-api-dev.abc-bg-api-dev.ApiGatewayRestApi.LogicalID.709]}
Ref: 
${Token[ApiGatewayRestApi.Ref.712]}
AttrRestApiID: 
${Token[TOKEN.710]}
AttrRootResourceID: 
${Token[TOKEN.711]}

for a Resource from a different stack (but triple-checked that it is instantiated on top of the above RestAPI) I got:

Ref:
${Token[ApiGatewayResourceA600n.Ref.632]} 
Path: 
"a600n" 
ParentID:${Token[TOKEN.627]}  <- this token should be equal to the one in RootResourceID of the RestAPI above.
APIGateway: ${Token[TOKEN.628]} <- and this token should be equal to the one in Ref of the RestAPI above.

And if I try to pull it from Cloudformation SDK as an Export (Output) I get this:
Export(ExportingStackId=arn:aws:cloudformation:us-east-1:xxxxx:stack/abc-bg-api-dev/a5a11930-8c9b-11eb-a230-0ad959ac0b5f, Name=dev-abcGWApiRestApiRootResourceId, Value=71cbfeeihd)

When I looked at what CDK synth output tells me (so stacks->CDK model->stacks), I see a link between the Resources and RestAPI made using an APIGatewayDeployment. But the Deployment has the same Fn:import just as every other element (resource, method) in the stack with resources using the APIGateway. I see nothing special in the Deployment. I even parsed the deployment and tried to pull something out of it with some "get..." functions. But there seems to be nothing useful.
Additionally, none of these allow me to use Stack.getTemplate().getOutput(LogicalID), as none of them seem to be a valid logicalID.

In summry: none of the things I tried allow me to state that the RestAPI is in fact the host of the Resource. I have been playing for 3 days with this and am starting to think there is magic involved. Can someone try to enlighten me please? Cheers.

[KKorzuno]

Hmm, nobody knows? :(