Universal bounded-tail conjecture for Biggs potentials of distance-regular graphs

Let $G$ be a distance-regular graph of diameter $D$, and let $\phi_0,\ldots,\phi_{D-1}$ denote its Biggs potentials. For an integer $m$ with $0\leq m\leq D-2$, the universal bounded-tail conjecture. There is a universal constant $K$ such that

$$\phi_{m+1}+\ldots+\phi_{D-1}\leq K\phi_m$$

for all $m$ and all distance-regular graphs. This conjecture seeks a uniform bound on the tail of the Biggs potentials relative to any preceding potential; the source gives a counterexample to the stronger bound with $K=1$, but does not establish whether some universal constant exists.