The band is too narrow. An Evidence Press briefing. The Mittag-Leffler function is a generalisation of the exponential function, with two parameters, called alpha and beta. It turns up wherever processes relax or wait in unusual ways: in fractional differential equations, in models of materials with memory, and in random processes that look the same at every scale. A useful question is when one over this function is completely monotone, meaning its derivatives alternate in sign. When that happens, the function is the Laplace transform of a probability distribution, which makes it far easier to use and to interpret. In 2025 Garrappa, Gerhold, Popolizio and Simon proved this holds when beta lies in a band from alpha minus one up to alpha, for alpha between one and two, and they conjectured that the band is exactly the right answer. This paper shows the band is too narrow, on both sides. The first result covers every alpha between one and two. Below the band, all the way down to zero, the reciprocal is still completely monotone; in fact its logarithm is a Bernstein function, so every power of it is a probability transform. The proof is short: it combines an identity hidden inside the earlier paper's own argument with a classical recurrence. The second result looks closely at alpha near two. There the exact set of good values of beta is an interval from zero up to a boundary b of alpha. At alpha equal to two the boundary is three, and its slope there is about two point eight three, given by an exact formula. The boundary is where two zeros of the function on the negative axis collide and leave the real line. Because the boundary sits above alpha, the band fails from above as well. What is not proved: how close to two alpha must be, and the full answer for other alpha, which is left as a conjecture supported by computation. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on the fourth of November twenty twenty-six, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.