Zeta-value conjecture for renewal sequence point probabilities

Let $C_k^{1,1}$ be the renewal-sequence random variable whose point probabilities are under consideration, and let $\zeta(j)$ denote the Riemann zeta function. For each $k\geq 1$ and $0\leq j\leq k$, write the point probability in terms of rational coefficients. Zeta-value conjecture. For each $k\geq 1$ and $0\leq j\leq k$,

$$\mathbb{P}(C^{1,1}_k = j) = q_{k,1} + \sum_{j = 2}^k q_{k,j} \zeta(j),$$

with $q_{k,j}$ rational numbers. This conjecture extends the observed formulas through $k=4$ and the cases $j=0,k-1,k$; an explicit formula for the point probabilities for general $j$ and $k$ remains open.