Suppose we have a Myerson-Satterthwaite model of bilateral trade between a buyer and seller. The buyer has private information about their valuation, $v$, and the seller has private information about their cost, $c$, of providing the good. There is a common prior that $v\sim F$ and $c\sim G$ for some smooth CDFs $F,G$ which are strictly increasing over intervals $[\underline{v},\bar{v}]$ and $[\underline{c},\bar{c}]$, respectively, and are flat outside of these intervals. Both agents have normalised reservation utilities of $0$. We say a mechanism $\mu$ is implementable if it is both incentive compatible and individually rational.
Define the most preferred implementable mechanisms of the buyer and seller as $\mu_B$ for the former and $\mu_S$ for the latter. These will be mechanisms corresponding to take-it-or-leave-it price offers. Define the random dictatorship (RD) mechanism with bargaining parameter $\beta$ as the one which selects $\mu_B$ with probability $\beta$ and $\mu_S$ with probability $1-\beta$. We write the RD($\beta$) mechanism in short as $\beta \mu_B +(1-\beta) \mu_S$. This mechanism is implementable.
Fix some $\beta\in(0,1)$. I am interested in whether there exists some implementable mechanism $\mu^*$ such that the interim utilities of both the buyer and seller are weakly larger under $\mu^*$ than the RD($\beta$) mechanism and for at least one type, the interim utility is strictly larger.
Effectively, I am asking whether the RD mechanism is incentive-efficient in the terminology of Myerson (1984; “Two-Person Bargaining Problems with Incomplete Information”). If it is not incentive-efficient, what is a good counterexample and what goes wrong with the RD mechanism?
Edit: The uniform case may provide a counterexample. If $v\sim [0,1]$ and $c\sim [0,1]$, then an example of a Myerson-Satterthwaite (1983) second-best efficient mechanism has allocation rule $p$ and transfer (from seller to buyer) function $x$ of the form $$\begin{align*} p(v,c) &= \begin{cases}1 & \text{ if } c\leq v-\frac{1}{4} \\ 0 & \text{o/w} \end{cases} \\ x(v,c) &= \begin{cases} \frac{v+c+0.5}{3}& \text{ if } c\leq v-\frac{1}{4} \\ 0 & \text{o/w} \end{cases} \end{align*}$$ This transfer rule corresponds to the Chatterjee-Samuelson bilateral auction. Clearly, this cannot be an interim efficiency improvement over the RD($\beta$) mechanisms since cost types $c\in (3/4,1]$ and valuation types $[0,1/4)$ never trade so are worse off in this new mechanism. However, I think there must be some way to adjust this transfer rule so as to distribute the efficiency gains in such a way that it is an interim efficiency improvement over the RD($\beta$) mechanisms.