Conjecture identifying affine $k$-Schur functions with $k$-atoms

For $k>0$, let $\mathcal P^k$ be the set of partitions with first part at most $k$, let $s_\lambda^{(k)}[X;t]$ be the $k$-Schur function with parameter $t$, and let $A_\lambda^{(k)}[X;t]$ denote the $k$-atom. Identification conjecture. For every $\mu\in\mathcal P^k$,

$$s_\mu^{(k)}[X;t]=A_\mu^{(k)}[X;t].$$

This conjecture connects the newly constructed $t$-generalization of $k$-Schur functions to the $k$-atoms of Lapointe, Lascoux, and Morse. The source introduces it after proving the basis and specialization properties of the new functions, but supplies no proof of the identification.