Essential positivity for Toeplitz operators on the Fock space

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Per\"al\"a and Virtanen. We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Per\"al\"a and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Per\"al\"a and Virtanen holds true, even without radiality.