In this note we show that the problem of distributed diagnosis under unbounded communication delay is decidable when there is no inferencing involved among the diagnosers. The notion of joint(infinity)-diagnosability is introduced to capture the diagnosability property in this setting. We show the equivalence of joint(infinity)-diagnosability and codiagnosability, which captures the diagnosability property in the decentralized setting (i.e., one involving no communication). Thus the decidability result follows from the decidability of codiagnosability established in a previous paper. We also show that the property of joint(infinity) -diagnosability is stronger than decentralized-diagnosability introduced in a previous paper.