Fermionic formula for double Kostka polynomials

The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{\Bla,\Bmu}(t),$ indexed by two double partitions $\Bla,\Bmu,$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{\Bla,\Bmu}(t)$ in the special case where $\Bmu=(-,\mu'').$ We formulate a $1D$ sum and a fermionic formula for $K_{\Bla,\Bmu}(t),$ as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the $X=M$ conjecture.