Explicit formulas conjecture for special symmetric-decomposition polynomials

Let $C_k$ denote the $k$th Catalan number, let $\operatorname{Nar}_{k}(t)$ denote the Narayana polynomial, and define

$$\overline{\operatorname{Nar}}_{k}^{(j)}(t)=\sum_{i=0}^{k-j}\frac{j+1}{k+1}\binom{k+1}{i}\binom{k+1}{i+j+1}t^{i}.$$

Let $W_{n,k}^{+}(t)$ and $W_{n,k}^{-}(t)$ be the symmetric-decomposition polynomials. Special-formula conjecture. For $k\geq1$,

$$W_{2k,k}^{+}(t)=(k-1)C_k\operatorname{Nar}_{k-1}(t),\qquad W_{2k,k}^{-}(t)=C_k\sum_{i=0}^{k-1}\binom{k-1}{i}^{2}t^i.$$

For $k\geq2$,

$$W_{2k,k+1}^{-}(t)=\binom{2k}{k}\operatorname{Nar}_{k-1}(t),\qquad W_{2k,k-1}^{+}(t)=-\frac{t}{2}\binom{2k}{k-2}\overline{\operatorname{Nar}}_{k-2}^{(1)}(t).$$

These formulas are intended to aid the real-rootedness classification; the source gives no proof and no formulas for the other symmetric-decomposition polynomials.