The modified log-Sobolev constant and the entropy contraction coefficient of a channel

Let $K$ be a channel, that is, a Markov kernel, from a finite set of at least two inputs to a finite set, and let $\mu$ be an input law of full support. The entropy contraction coefficient, or strong data processing constant, $\eta_{KL}$ is the smallest $\eta$ with $D(\nu K \parallel \mu K) \le \eta D(\nu \parallel \mu)$ for every input law $\nu$, where $D$ is the Kullback-Leibler divergence. Let $K^*$ be the reverse channel, which gives the posterior law of the input given an output when the input has law $\mu$. One step of the chain with kernel $KK^*$ on the inputs goes from an input to an output drawn from $K$ and then to an input drawn from $K^*$. The kernel $KK^*$ is reversible with respect to $\mu$, and $\rho_0$ denotes its modified log-Sobolev constant, the largest rate of exponential entropy decay of the continuous-time chain with generator $KK^* - I$. Raginsky (2016) stated, with a proof, that $\rho_0 \le 1 - c\eta_{KL}$ for a universal constant $c > 0$. Caputo, Chen, Gu and Polyanskiy (2025) showed that his proof is incorrect and left the bound open. The bound is false, because it fails on Raginsky's own example, in which a channel flips a uniform binary input with a fixed probability, the crossover probability. At crossover probability $1/2$ the output law is the same for every input, so $\rho_0 = 2$ and $\eta_{KL} = 0$, which violates the bound for every $c$. A bound of this form therefore needs the constant $2$ in place of $1$. We prove $\rho_0 + 0.16\eta_{KL} \le 2$ for every channel between finite sets and every input law of full support on at least two points, and $\rho_0 + \eta_{KL} \le 2$ when, in addition, the input set has two points. Both proofs certify some steps by interval arithmetic, which encloses each computed quantity in an interval that is guaranteed to contain its exact value. On two points, we express the supremum of $\rho_0 + \eta_{KL}$ over all channels through the input law and a one-dimensional minimization, and show that the supremum equals $2$ only for the uniform law. We conjecture that $\rho_0 + \eta_{KL} \le 2$ for every channel between finite sets and every input law of full support on at least two points.