Conjectured non-closure for incompatible signatures in nondeterministic input-driven PDA with translucent letters

Establish that, without assuming compatible signatures, the family L(NIDPDAwtl) is not closed under union and the family L(nrNIDPDAwtl) is not closed under union and intersection.

Background

The authors point out that union closure holds for both nondeterministic families when signatures are compatible. Beyond this special case, they state a strong conjecture of non-closure, covering the open scenarios for union (both families) and intersection (non-returning family).

This conjecture aims to characterize the closure behavior in the general, more natural setting where the signature partitions may differ across devices.