SSM parameters SAM local aws set up: unable to set properties to work in local - aws-lambda

Question

people.
I have been trying to set up a project to test my lambdas in a local environment. We are using sam to mimic AWS resources and everything works pretty fine with one exception: for business reasons, I had to include SSM parameters. When we try to read the parameters using SAM with sam local start-lambda ... whatever, the lambda code is not able to recover the parameters. I know that AWS credentials are fine as I can connect to AWS, but we don't want to use real AWS services to do that, we want to pass, set, define; whatever you want to call it is fine, SSM parameters in my local environment and then use them with sam local start-lambda whatever, without AWS connection therefore we can use parameters for testing only in local.
I have read the following post, How to access SSM Parameter Store from SAM lambda local in node
And this issue in github:
https://github.com/aws/aws-sam-cli/issues/616
It is mentioned that the way to do it is by using --env-vars but it is not working so far.
This is my template.
Parameters:
IdentityNameParameter:
Type: AWS::SSM::Parameter::Value<String>
Resources:
MyLambdaFunction:
Type: AWS::Serverless::Function
Properties:
Handler: handler::handleRequest
CodeUri: lambdaFunction
Runtime: java11
Timeout: 40
Environment:
Variables:
AWS_ACCESS_KEY_ID: "keyid"
AWS_SECRET_ACCESS_KEY: "accesskey"
AWS_DEFAULT_REGION: "us-west-1"
AWS_REGION: "us-west-1"
This is what I use to start the lambda
sam local start-lambda --host 0.0.0.0 -d 5859 --docker-volume-basedir /folderWithClasses --container-host host.docker.internal --debug --env-vars env.json
This is the env.json:
{
"Parameters": {
"IdentityNameParameter": "admin"
}
}
I guess there is no support to do this. If so, what's the point of having SAM to test it in local if you need AWS to actually test?
Any clue?