Case $\boldsymbol i$. If $lf$ is part of an optimal matching, and $v_{lf}$ increases, it remains so. (Otherwise there is a matching which pays off more than the original optimal match, does not match $lf$, and so would have been feasible and have the same payoff at the initial constraints.) The set of dual solutions increases the payoffs available to $lf$, and leaves everything else unchanged. In paticular the minimum wage to $l$ and the minimum profit to $f$ do not change.

Take any dual solution to the new problem. Every other pair must be dividing up the value of their match, so the set of allocations of these surpluses in the old and new problem must be identical. And $lf$ must divide their surplus $v'_{lf}$, so their payoff set has strongly increased.

Case $\boldsymbol{ii}$. $lf$ is out-of-the-money. Let $\sigma(l)$ denote $l$s optimal match. Suppose $v'_{lf}>v_{lf}$, $\sigma(l)\neq f$ and $\sigma$ is optimal on the interval $[v_{lf},v'_{lf}]$, ceteris paribus. Consider the minimal wage for $l$ and the maximal profit for $\sigma(l)$: \begin{equation*} \underline{w}_l+\bar\pi_{\sigma(l)}=v_{l\sigma(l)}. \end{equation*}