The Naisargik mapping between quaternary and binary Helberg codes

Let $H(n,q,s,a)$ denote a Helberg code with length $n$, alphabet size $q$, deletion parameter $s$, and residue $a$. Let $a$ be the residue corresponding to the maximum number of codewords in $H(n,4,1,a)$, and let $a'$ be the corresponding binary-code residue. Naisargik mapping conjecture. There is a one-to-one mapping $\phi$ between the codewords of $H(n,4,1,a)$ and $H(2n,2,2,a')$ such that, for every codeword $c\in H(n,4,1,a)$, $\phi(c)\in H(2n,2,2,a')$, and

$$\phi(H(n,4,1,a))=H(2n,2,2,a').$$

Thus all codewords of the Naisargik image satisfy $\phi(H(n,4,1,a))\subseteq H(2n,2,2,a')$. The mapping is used in the paper's argument relating deletion correction in quaternary Helberg codes to correction in their binary Naisargik images; the supplied text gives no independent resolution evidence for this assertion.