The conditional-convergence conjecture for coefficients of L-functions

Let $L(s)=\sum b(n)/n^s$ be an entire $L$-function of degree $k$, normalized so that its critical line is $\operatorname{Re}(s)=1/2$. Let $P_L$ denote a polynomial. More generally, suppose that $L(s)$ is meromorphic and has only a pole at $s=1$, of order $r$.

Conditional-convergence conjecture. For an entire $L$-function,

$$\sum_{n\leq X}b(n)=O\left(X^{(k-1)/2k+\epsilon}\right).$$

More generally, for a meromorphic $L$-function as above,

$$\sum_{n\leq X}b(n)=XP_L(\log X)+O\left(X^{(k-1)/2k+\epsilon}\right),$$

where $P_L$ is a polynomial of degree $r-1$.

This is proposed by analogy with the conjectured remainder size in the $k$-divisor problem: after removing the polynomial main term caused by the pole, the coefficients should exhibit cancellation of order $X^{(k-1)/2k+\epsilon}$.