Logarithmic upper bound for the Nnamlerinchs constant

For a fixed base $b\geq 2$, let $\overleftarrow{K}_b$ be the smallest number greater than $1$ such that every sufficiently large integer is a sum of at most $\overleftarrow{K}_b$ reversed primes in base $b$. Nnamlerinchs constant upper-bound conjecture. For every $b\geq 2$,

$$\overleftarrow{K}_b\ll\log b.$$

This conjecture strengthens the existence conjecture by proposing a uniform logarithmic upper bound. The paper proves that on an infinite increasing sequence of bases, the constants have a matching lower bound $\overleftarrow{K}_{b_i}\gg\log b_i$, so the predicted order would be sharp along that sequence.