= Solution
The <law of iterated expectation> yields
$$
\mathbb E[p_T\mid p_M]=\frac{\alpha+n p_M}{s+n},\qquad
\operatorname{Cov}(p_M,p_T)=\frac n{s+n}v.
$$
One may obtain $\operatorname{Var}(p_T)=v$ either from its unchanged <marginal distribution> proved below or directly from the <law of total variance>: the variance of its conditional mean is $nv/(s+n)$ and its expected conditional variance is $sv/(s+n)$. Hence the <correlation coefficient> is exactly
$$
\boxed{\operatorname{Corr}(p_M,p_T)=\frac n{n+\alpha+\beta}\longrightarrow1.}
$$
Also $\mathbb E[(p_T-p_M)^2]=2v s/(s+n)\to0$. \b[The two probabilities become close while keeping their original beta marginals.]
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