Existence of the Nnamlerinchs constant

For a fixed base $b\geq 2$, define the Nnamlerinchs constant $\overleftarrow{K}_b$ to be the smallest number greater than $1$ such that every sufficiently large integer can be written as the sum of at most $\overleftarrow{K}_b$ reversed primes in base $b$. Nnamlerinchs constant existence conjecture. For every $b\geq 2$,

$$\overleftarrow{K}_b<\infty.$$

The paper notes that this is much weaker than the pure reversed-prime representation conjecture, but it is not known whether the constant is finite for any general base. A theorem in the paper shows that along an infinite sequence of bases the constants, when defined, have lower bounds of order $\log b$.