The affine character conjecture for the $F_k(q)$ series

Let $F_k(q)$ be defined by

$$\sum_{n_1,n_2,\dotsc,n_{2k+1} \geq 0 } \frac{q^{\sum_{i=1}^{2k+1} n_i^2-\sum_{i=1}^{2k} n_{i} n_{i+1} }} {(q)_{n_1}^2 (q)_{n_2}^2 \cdots (q)_{n_{2k+1}}^2} = \frac{1}{(q)_\infty^{2k+1}} F_k(q).$$

Here $k\geq 1$, and $L_{\mathfrak{sp}(2k)}(\Lambda_0)$ denotes the suitably normalized level-one affine vertex algebra of type $C^{(1)}_k$, with character $\operatorname{ch}[L_{\mathfrak{sp}(2k)}(\Lambda_0)](q)$. The affine character conjecture. For every $k\geq 1$,

$$F_k(q)=\operatorname{ch}[L_{\mathfrak{sp}(2k)}(\Lambda_0)](q).$$

This predicts a family of $q$-series identities relating the series $F_k(q)$ to characters of level-one affine vertex algebras of type $C^{(1)}_k$.