  We consider the least singular value of $M = R^* X T + U^* YV$, where
$R,T,U,V$ are independent Haar-distributed unitary matrices and $X, Y$ are
deterministic diagonal matrices. Under weak conditions on $X$ and $Y$, we show
that the limiting distribution of the least singular value of $M$, suitably
rescaled, is the same as the limiting distribution for the least singular value
of a matrix of i.i.d. gaussian random variables. Our proof is based on the
dynamical method used by Che and Landon to study the local spectral statistics
of sums of Hermitian matrices.
