Charge-and-tightness conjecture for the $R^p$ ropelength energies

Let $L$ be a link, let $R^p(L)$ be the scale-invariant energy defined in the paper, and let $R(L)$ denote ropelength, with thickness measuring the radius of the largest embedded tube around $L$. An energy is charge and tight when it diverges along sequences approaching a self-intersection or a shrinking knotted arc. Charge-and-tightness conjecture. For $p\geq 2$, the energy $R^p$ is bounded below by some monotonic function of ropelength, so $R^p$ approaches infinity for any sequence of links with fixed length and thickness approaching zero. This conjecture would establish the stated charge and tight properties for the energies with $p\geq 2$; the paper does not provide a resolution.