KPZ equation tails for general initial data

We consider the upper and lower tail probabilities for the centered (by time$\mathrm{/}24$) and scaled (according to KPZ time${}^{1\mathrm{/}3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5\mathrm{/}2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3\mathrm{/}2$ at all depth in the tail.