The critical Schatten exponent for the Berger-Coburn endpoint problem

Berger and Coburn showed that boundedness of a Toeplitz operator $T_g$ on the Fock space controls the heat transform $g^{(t)}$ for $1/4<t<1$, and conjectured the endpoint $g^{(1/4)}$ characterizes boundedness. Looi recently disproved this by constructing a bounded $T_g$ with unbounded $g^{(1/4)}$. We show that $p=1$ is the exact Schatten exponent for which $T_g\in S_p$ forces $g^{(1/4)}$ to be bounded. The trace-class operators appearing in Berger and Coburn's trace formula have divergent trace norms as $t\downarrow1/4$, yet converge strongly to $2^n J$, and it follows that $$ g^{(1/4)}(a)=2^n\operatorname{tr} \bigl(T_gW_aJW_a^*\bigr),\qquad \|g^{(1/4)}\|_\infty\leq2^n\norm{T_g}_{S_1}, $$ for every admissible symbol $g$ with $T_g$ trace class, where $J$ is the parity operator and $W_a$ is Weyl translation. We prove that $2^n$ is optimal, and the same bound holds for $T_g\in S_p$ with $0<p\leq1$. For $p>1$, no corresponding $S_p$ estimate is possible, even for compactly supported smooth symbols. Moreover, a Baire category argument shows there is an admissible symbol $g$ with $T_g\in S_p$ for every $p>1$ while $g^{(1/4)}$ is unbounded, though it yields no explicit symbol. We also determine the optimal constants in the Berger--Coburn estimates for $1/4<t\leq1$.