The paper—bright side of supplier encroachment—is well write. Here are key points.

R’s selling advantage stems from 1) superior knowledge, 2) direct contact, 3) economies of scope.

The direct channel is the centralized system. The selling cost $c\in R_+$ regulates/allocates the outputs through each channel.

The changes in $w$ and $q$ are means to achieve the goal of profit maximization. Therefore, the ultimate determinant is the profitability, not the means themselves. I suspect the profit per unit is the same for both channels: $w^E= a/2 - c/6$. For the direct channel, it is $(P-c)=a/2-5c/6$.

I don’t find the wholesale price logic compelling. Here is my take. In the dual channel setting, M must allocate supply. The allocations depend on several factors: (1) the end demand determines the total supply; (2) $q_M$ is direct, but $q_R$ is indirectly determined by $w$ in the form of the best response (BR) function $q_R^E(w) = (a+ c - 2w)/2b$.

So the key is to compare their BRs, which summarize R’s strategic calculation:
$q_R^N(w)= (a-w)/2b,$
$q_R^E(w) = (a+ c - 2w)/2b.$
Because of the encroachment, the game structure changes; so do the best response functions. This is the key; the wholesale price change is secondary.