= Solution
Let $v=\operatorname{Var}(S)>0$ and $v_I=\operatorname{Var}(S_I)$. The variance-matching <quota share reinsurance> in this part has the nonnegative retained fraction $\alpha^*=\sqrt{v_I/v}$. For claimwise retentions $0\leq h(x)\leq x$, its existence within $[0,1]$ follows from $v_I=\lambda\mathbb E[h(X)^2]\leq\lambda\mathbb E[X^2]=v$. The comparison below only needs the matching contract specified in the question.
Because $S_R=S-S_I$, the <variance> and <covariance> identities give
$$
\operatorname{Var}(S_I)+\operatorname{Var}(S_R)
=v+2v_I-2\operatorname{Cov}(S,S_I).
$$
The <Cauchy-Schwarz inequality> bounds $\operatorname{Cov}(S,S_I)\leq\sqrt{vv_I}=\alpha^*v$. Therefore the <quota share comparison at matched retained variance> is
$$
\operatorname{Var}(S_I)+\operatorname{Var}(S_R)
\geq v+2(\alpha^*)^2v-2\alpha^*v
=\bigl((\alpha^*)^2+(1-\alpha^*)^2\bigr)v.
$$
The last expression is the sum of <variances> under the matching <quota share reinsurance>. Hence
$$
\boxed{\operatorname{Var}(S_I^*)+\operatorname{Var}(S_R^*)
\leq\operatorname{Var}(S_I)+\operatorname{Var}(S_R).}
$$
If $v=0$, the given matching contract forces $v_I=0$, so all relevant totals are constant and equality holds. For $v>0$, equality in the bound is equivalent to $S_I-\mathbb E[S_I]=\alpha^*(S-\mathbb E[S])$ almost surely.
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