Without-replacement matrix-product norm conjectures

Let $[n]=\{1,\ldots,n\}$, let $\bm{A}_1,\ldots,\bm{A}_n$ be positive semidefinite matrices, and let $k$ be a positive integer. For a function of an ordered $k$-tuple, define

$$\operatorname{\mathbb{E}}_{\mathrm{wo}}[f(x_{i_1},\ldots,x_{i_k})]=\frac{(n-k)!}{n!}\sum_{j_1\neq j_2\neq\cdots\neq j_k}f(x_{j_1},\ldots,x_{j_k}),$$

and

$$\operatorname{\mathbb{E}}_{\mathrm{wr}}[f(x_{i_1},\ldots,x_{i_k})]=n^{-k}\sum_{(j_1,\ldots,j_k)=1}^n f(x_{j_1},\ldots,x_{j_k}).$$

Here $\operatorname{\mathbb{E}}_{\mathrm{wo}}$ and $\operatorname{\mathbb{E}}_{\mathrm{wr}}$ denote without-replacement and with-replacement expectations, respectively. Without-replacement matrix-product norm conjectures. The following two inequalities always hold:

$$\left\|\operatorname{\mathbb{E}}_{\mathrm{wo}}\left[\prod_{j=1}^k\bm{A}_{i_j}\right]\right\|\leq\left\|\operatorname{\mathbb{E}}_{\mathrm{wr}}\left[\prod_{j=1}^k\bm{A}_{i_j}\right]\right\|$$

and

$$\left\|\operatorname{\mathbb{E}}_{\mathrm{wo}}\left[\prod_{j=1}^k\bm{A}_{i_{k-j+1}}\prod_{j=1}^k\bm{A}_{i_j}\right]\right\|\leq\left\|\operatorname{\mathbb{E}}_{\mathrm{wr}}\left[\prod_{j=1}^k\bm{A}_{i_{k-j+1}}\prod_{j=1}^k\bm{A}_{i_j}\right]\right\|.$$

These inequalities are proposed as sufficient conditions for proving that without-replacement sampling outperforms with-replacement sampling in randomized iterative methods. The supplied text gives no resolution of either inequality.