The shifted moments conjecture for Dirichlet $L$-functions

Let $q$ tend to infinity through primes, let $Y$ be the shift-height parameter, and let $R$ be a piecewise smooth positively oriented path encircling $0$ and $-s$. Dirichlet shifted moments conjecture. For $|\Re z|,|\Re s|\leq3/\log q$ and $|\Im z|,|\Im s|\ll Y$,

$$\frac1{\varphi(q)}\sum_{\chi\bmod q}L\left(\frac12+s-z,\chi\right)^kL\left(\frac12+z,\overline\chi\right)^k=\frac{a_k}{k!(2\pi i)^k}\int_{R^k}q^{\sum_{j=1}^k w_j}\prod_{i=1}^k w_i^{-k}(w_i+s)^{-k}\prod_{i\ne j}(w_i-w_j)\,d\boldsymbol w+O_k\left((\log q)^{k^2-1+\varepsilon}(1+\min(|s|,1)\log q)^{-k^2/2+1}\right).$$

This is the Dirichlet-$L$ analogue of the shifted zeta moments conjecture and is used to derive the arithmetic-progression variance formula conditionally. The full conjecture remains open.