Nearest southeast submatrix that makes multiple a prescribed eigenvalue. Part 1

Given four complex matrices A,B,CA,B,C and D  , where A∈Cn×nA∈Cn×n and D∈Cm×mD∈Cm×m, and given a complex number z0z0: What is the (spectral norm) distance from D   to the set of matrices X∈Cm×mX∈Cm×m such that z0z0 is a multiple eigenvalue of the matrixABCX?This problem is solved when z0z0 is not an eigenvalue of A  . We also give a conjecture when z0z0 is an eigenvalue of A.