Signed mean value trace inequalities for Hermitian matrices

Let $\mathsfsl{A},\mathsfsl{B},\mathsfsl{C}\in\mathbb{H}^{d}$ be Hermitian matrices, let $q$ be a positive integer, and let $s>0$. For a Hermitian matrix $\mathsfsl{X}$, write $\mathsfsl{X}_+$ and $\mathsfsl{X}_-$ for its positive and negative parts, and write $|\mathsfsl{X}|$ for its absolute value. Signed mean value trace inequalities. The following inequalities should hold:

$$\operatorname{tr}\big[\mathsfsl{C}(\mathrm{e}^{\mathsfsl{A}}-\mathrm{e}^{\mathsfsl{B}})\big]\leq \frac{1}{2}\operatorname{tr}\big[((s(\mathsfsl{A}-\mathsfsl{B})_+^2+s^{-1}\mathsfsl{C}_+^2)\mathrm{e}^{\mathsfsl{A}}+(s(\mathsfsl{A}-\mathsfsl{B})_-^2+s^{-1}\mathsfsl{C}_-^2)\mathrm{e}^{\mathsfsl{B}})\big]$$

and

$$\operatorname{tr}\big[\mathsfsl{C}(\mathsfsl{A}^{q}-\mathsfsl{B}^{q})\big]\leq \frac{q}{2}\operatorname{tr}\big[((s(\mathsfsl{A}-\mathsfsl{B})_+^2+s^{-1}\mathsfsl{C}_+^2)|\mathsfsl{A}|^{q-1}+(s(\mathsfsl{A}-\mathsfsl{B})_-^2+s^{-1}\mathsfsl{C}_-^2)|\mathsfsl{B}|^{q-1})\big].$$

These inequalities would provide the missing ingredient for extending the paper's concentration method to matrix self-bounding functions. The source presents them as needed for that analysis, but gives no resolution, so their status remains open.