Type-C q-multiplicity and one-dimensional-sum identities

Let $\mathcal{P}_n$ be the set of partitions with the relevant type-$C_n$ restrictions, and let $u_{\lambda,\mu}(q)$, $U_{\lambda,\mu}(q)$, and $X_{\lambda,\mu}(q)$ be the $q$-multiplicity and one-dimensional-sum polynomials defined from the type-$C_n^{(1)}$ crystals. For $\lambda\in\mathcal{P}_n$, the source proposes the following identities:

Type-C identities.

$$u_{\lambda,(1^n)}(q)=q^{\frac{|\mu|-|\lambda|}{2}}X_{\lambda,(1^n)}(q),$$

and

$$U_{\lambda,\mu}(q)=q^{|\mu|-|\lambda|}X_{\lambda,\mu}(q)\qquad\text{for any }\mu\in\mathcal{P}_n.$$

These identities are motivated by computations and by the agreement of the type-$A$ and type-$C$ energy functions in the equal-size case; the source does not state a resolution of them.