Complex almost-injective intensity measurements have NP-complete consistency

Let $\mathcal{F}=\{\Phi_M\}_{M=2}^\infty$ be a family of ensembles $\Phi_M=\{\varphi_{M;n}\}_{n=1}^{2M}\subseteq\mathbb{C}^M$ which yield almost injective intensity measurements and have complex rational entries that can be computed in polynomial time. The consistency conjecture. Then $\textsc{ConsistentIntensities}[\mathcal{F}]$ is $\NP$-complete. This would strengthen the preceding hardness result for the smallest possible almost-injective ensembles in the complex case, paralleling the established result for full spark ensembles with $M+1$ measurements; the conjectured NP-completeness is not resolved in the supplied text.