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Claim. For $1$-bounded multiplicative functions put $\mathbb D(f,g;x)^2=\sum_{p\le x}(1-\operatorname{Re}f(p)\overline{g(p)})/p$, and write $\mathbb E^{\log}_{n\le X}$ for the logarithmic average over $n\le X$. Assume the three-function case of the logarithmically averaged Elliott conjecture, in the form of Conjecture 1.3(ii) of Tao and Teräväinen's 2019 structure paper: if $g_1,g_2,g_3$ are $1$-bounded multiplicative functions and one of them satisfies $\inf_{|t|\le X}\mathbb D(g_j,\chi(n)n^{it};X)\to\infty$ as $X\to\infty$ for every Dirichlet character $\chi$, then $\mathbb E^{\log}_{n\le X}g_1(n)g_2(n+1)g_3(n+2)\to0$. Then $\Lambda(k,3)<\infty$ for every odd $k$. Under the same assumption, every completely multiplicative $f:\mathbb N\to\mu_k$ with $k$ odd takes the value $1$ at $n$, $n+1$ and $n+2$ for some $n$. The step from such functions to $\Lambda(k,3)$ is unconditional: for odd $k$, $\Lambda(k,3)$ equals the supremum, over all completely multiplicative $f:\mathbb N\to\mu_k$, of the least $n$ with $f(n)=f(n+1)=f(n+2)=1$, taken as $\infty$ when there is none.

Covers. The second question of Problem 436, whether $\Lambda(k,3)$ is finite for every odd $k$, answered yes conditionally on the assumption above, which is an open conjecture. Unconditionally the claim adds no case: $k=3$ is settled by the accepted Lehmer, Lehmer, Mills and Selfridge claim, and every odd $k\ge5$ remains open. No bound on $\Lambda(k,3)$ is given, so the growth question is not addressed, nor is the first question.

Credit. The assumption is Tao and Teräväinen's conjecture, recorded with their 2019 structure paper. The proof uses the Chebotarev density theorem, the fixed-modulus Vinogradov–Korobov bound for Dirichlet $L$-functions, the fixed-modulus prime number theorem in arithmetic progressions with Mertens' estimates, and the fundamental lemma of sieve theory, all at their published statements. The realization of prescribed character values at primes, the identity for $\Lambda(k,3)$, the sieve argument for functions at bounded pretentious distance from a Dirichlet character, and the conditional deduction are Plasma AI's.

Standing. Author-recorded at tier 0, with no independent review and no executable evidence. The assumption is unproved, so the claim derives nothing for the problem's standing. Catalog acceptance is pending.

Depends on. The odd-order triple lead (tier 0), through Sections 1, 4 and 7 of its proof sketch.