  We give the first super-polynomial (in fact, mildly exponential) lower bounds
for tolerant testing (equivalently, distance estimation) of monotonicity,
unateness, and juntas with a constant separation between the "yes" and "no"
cases. Specifically, we give
  $\bullet$ A $2^{\Omega(n^{1/4}/\sqrt{\varepsilon})}$-query lower bound for
non-adaptive, two-sided tolerant monotonicity testers and unateness testers
when the "gap" parameter $\varepsilon_2-\varepsilon_1$ is equal to
$\varepsilon$, for any $\varepsilon \geq 1/\sqrt{n}$;
  $\bullet$ A $2^{\Omega(k^{1/2})}$-query lower bound for non-adaptive,
two-sided tolerant junta testers when the gap parameter is an absolute
constant.
  In the constant-gap regime no non-trivial prior lower bound was known for
monotonicity, the best prior lower bound known for unateness was
$\tilde{\Omega}(n^{3/2})$ queries, and the best prior lower bound known for
juntas was $\mathrm{poly}(k)$ queries.
