Let $p$ and $q$ be two probability measures on some measure space $(X,\mathcal X)$. Suppose that there is $\lambda>0$ such that $(1+\lambda) p-\lambda q\geqq 0$, in other words there is $\gamma\in]0,1[$ and $q'$ a probability measure such that $p=\gamma q+(1-\gamma)q'$. For any $A\in\mathcal X$ such that $p(A)>0$, let $p_A(\,\cdot\,)=\frac{p(A\cap \,\cdot\,)}{p(A)}$ and similarly for $q_A$. Observe that we get $p_A=\gamma q_A + (1-\gamma)q_A'$. Let $B\in\mathcal X$ be the Hahn decomposition of $p-q$, i.e. for any $C\in\mathcal X$, $(p-q)(B\cap C)\geq 0$ and $(p-q)(\overline B\cap C)\geq 0$. Assume that $p(B), q(B)\in]0,1[$, indeed otherwise $p=q$, this is because the total variation between $p$ and $q$ is exactly $p(B)-q(B)$.
I am wondering if the following type of bound is possible
There exists a constant $a(\lambda)<1$ such that $$\sup_{C\in\mathcal X}q(B)(p_B(C)-q_B(C))+q(\overline B)(p_{\overline B}(C)-q_{\overline B}(C))\leq a(\lambda) (p(B) - q(B))$$
Observe that this is essentially comparing the total variation of $q(B)(p_B-q_B)+q(\overline B)(p_{\overline B}-q_{\overline B})$ and that of $p-q$. Since $p_B-q_B$ and $p_{\overline B}-q_{\overline B}$ have disjoint support and the optimization over C can be partitioned in the optimization over $B\cap C$ and $\overline B\cap C$, therefore \begin{align*} \| q(B)(p_B-q_B)+q(\overline B)(p_{\overline B}-q_{\overline B}) \|_{\text{TV}} &= q(B)\| p_B-q_B\|_{\text{TV}}+q(\overline B)\|p_{\overline B}-q_{\overline B}) \|_{\text{TV}} \end{align*}