Conjecture on parametric Euler sums in terms of Hurwitz zeta values

For positive integers $k$ and $m$, and complex $\alpha\in\mathbb{C}\backslash\mathbb{N}^-$, define the finite multiple harmonic sum $\zeta_{n-1}(\{1\}_m)$ by the notation used for multiple zeta sums. Consider the parametric Euler sums

$$\displaystyle\!\sum_{n=1}^\infty \displaystyle\!\frac{\zeta_{n-1}(\{1\}_m)}{n(n+\alpha)},\qquad \displaystyle\!\sum_{n=1}^\infty \displaystyle\!\frac{\zeta_{n-1}(\{1\}_m)}{(n+\alpha)^{k+1}}.$$

Parametric Euler-sum conjecture. These sums can be evaluated in terms of Hurwitz zeta values and digamma functions. This extends the preceding explicitly known cases, but the source does not provide the asserted evaluation formulas.