Let $X_i\in\{0,1\}$ be distributed according to $\mathrm{Bernoulli}(p_i)$ and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$, for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=1}^n X_i$ and $Y:=\sum_{i=1}^n Y_i$; both take values in $\{0,\ldots,n\}$.

Assume $p_i\ge q_i$ and even $n$, put $\gamma:=\frac1n\sum_{i=1}^n(p_i-q_i)$, and let $\bar X\sim\mathrm{Binomial}(n,1/2+ \gamma/2)$ and $\bar Y\sim\mathrm{Binomial}(n,1/2- \gamma/2)$.

Conjecture: $$ \max_{k\ge 0} | \mathbb{P}(X\ge k) - \mathbb{P}(Y\ge k) | \ge \mathbb{P}(\bar X\ge n/2) - \mathbb{P}(\bar Y\ge n/2) .$$

This would imply, in particular, a positive resolution of the conjecture in Minimizing total variation under constraint

Let $X_i\in\{0,1\}$ be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$ and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$, for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=1}^n X_i$ and $Y:=\sum_{i=1}^n Y_i$; both take values in $\{0,\ldots,n\}$.

Assume $p_i\ge q_i$ and even $n$, put $\gamma:=\frac1n\sum_{i=1}^n(p_i-q_i)$, and let $\bar X\sim\mathrm{Binomial}(n,1/2+ \gamma/2)$ and $\bar Y\sim\mathrm{Binomial}(n,1/2- \gamma/2)$.

Conjecture: $$ \max_{k\ge 0} | \mathbb{P}(X\ge k) - \mathbb{P}(Y\ge k) | \ge \mathbb{P}(\bar X\ge n/2) - \mathbb{P}(\bar Y\ge n/2) .$$

This would imply, in particular, a positive resolution of the conjecture in Minimizing total variation under constraint

Let $X_i\in\{0,1\}$ be distributed according to $\mathrm{Bernoulli}(p_i)$ and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$, for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=1}^n X_i$ and $Y:=\sum_{i=1}^n Y_i$; both take values in $\{0,\ldots,n\}$.

Assume $p_i\ge q_i$ and even $n$, put $\gamma:=\frac1n\sum_{i=1}^n(p_i-q_i)$, and let $\bar X\sim\mathrm{Binomial}(n,1/2+ \gamma/2)$ and $\bar Y\sim\mathrm{Binomial}(n,1/2- \gamma/2)$.

Conjecture: $$ \max_{k\ge 0} | \mathbb{P}(X> k) - \mathbb{P}(Y> k) | \ge \mathbb{P}(\bar X\ge n/2) - \mathbb{P}(\bar Y\ge n/2) .$$

This would imply, in particular, a positive resolution of the conjecture in Minimizing total variation under constraint

Let $X_i\in\{0,1\}$ be distributed according to $\mathrm{Bernoulli}(p_i)$ and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$, for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=1}^n X_i$ and $Y:=\sum_{i=1}^n Y_i$; both take values in $\{0,\ldots,n\}$.

Assume $p_i\ge q_i$ and even $n$, put $\gamma:=\frac1n\sum_{i=1}^n(p_i-q_i)$, and let $\bar X\sim\mathrm{Binomial}(n,1/2+ \gamma/2)$ and $\bar Y\sim\mathrm{Binomial}(n,1/2- \gamma/2)$.

Conjecture: $$ \max_{k\ge 0} | \mathbb{P}(X\ge k) - \mathbb{P}(Y\ge k) | \ge \mathbb{P}(\bar X\ge n/2) - \mathbb{P}(\bar Y\ge n/2) .$$

This would imply, in particular, a positive resolution of the conjecture in Minimizing total variation under constraint