Swiatak's second-derivative spanning conjecture for functional equations

Let $x\in R^n$ and $t\in R^r$, let $\Omega\subseteq R^r$ be open, and consider distribution solutions of the functional equation (1.1) satisfying assumptions A-1 through A-5. Let $t^\circ\in\Omega$ satisfy $\boldsymbol{\phi}_j(t^\circ)=0$ and $a_j(x,t^\circ)>0$ for all $x\in R^n$ and $j=1,2,\ldots,k$. A continuous solution is one satisfying conclusion [B] if it is $C^\infty$, and a locally integrable solution is one satisfying [B] if it is $C^\infty$ almost everywhere. Swiatak's conjecture. If assumption (3) of Swįatak's theorem is replaced by

$$\left\{\boldsymbol{\phi}_j^{\prime\prime}(t^\circ)\right\},\quad j=1,2,\ldots,k,$$

spanning $R^n$, then conclusion [B] may be valid. The conjecture proposes that second-order variation of the shifts can replace the first-derivative spanning condition in the hypoellipticity theorem; the supplied text gives no resolution.