PL Recognition After Two $S^2\times S^2$ Stabilisations is PSPACE-Hard

Let $Z$ be any fixed closed connected PL $4$-manifold. We give a polynomial-time many-to-one reduction from the compressed word problem in Thompson's group $F$ to fixed-target recognition of $Z\#_{2}(S^2\times S^2)$. Once a finite triangulation of $Z$ has been fixed, the construction sends a straigh