Create a Service and Implement a Caller.
Create an Apex class that is a service that exposes a bulkified service method. The service closes and sets the reason for one or more given case records. Implement an Apex REST class that calls this service.

My Service class is the following:
 

global with sharing class CaseService {

    global static void closeCases(List<Id> caseIds, String closingReason) {
        if (caseIds == null || caseIds.size() == 0) {
            throw new CaseServiceException('You should tell me which cases you wanna close');
        }
        
        if (closingReason == null || closingReason == '') {
            throw new CaseServiceException('I won\'t close a case without any reason');
        }
        
        List<Case> closedCases = new List<Case>();
        
        for(Id caseId : caseIds) {
            Case caseToClose = new Case(
            	Id = caseId,
                Status = 'Closed',
                Reason = closingReason
            );
            closedCases.add(caseToClose);
        }
        
        SavePoint sp = Database.setSavepoint();
        if (closedCases.size() > 0) {
            try {
                update closedCases;
            } catch (Exception e) {
                Database.rollback(sp);
                throw e;
            }
        }
    }
    
    public class CaseServiceException extends Exception {}
}

And my webservice endpoint (I can't really call it restful) is :
 

@RestResource(urlMapping='/case/*/close')
global with sharing class CaseCloseResource {
    
    @HttpPost
    global static void closeCase(String reason) {                
        Id id = RestContext.request.requestURI.substringBetween('case/', '/close');
        CaseService.closeCases(new Id[]{id}, reason);
    }
}

I tried in an anonymous code execution :
 

Id id = '5000Y000002Vc8Y';
CaseService.closeCases(new Id[]{id}, 'You shall be closed!');

And successfully got the following : 
Then, using workbeng, I tried the following request :

POST /services/apexrest/case/5000Y000002Vc8m/close
Payload : {"reason" : "Please close this case too!"}

And got a 200 response from the server as well as 
That's why I am still confused on why the challenge checker is rejecting me ?
Did I miss something the challenge requested me to do ?