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Fifty nine words place I second and M third, and one opening letter takes far more than its share.
L takes the first box in seventeen of the fifty nine, well ahead of M and T on six each. LIMIT, LIMBS, LIMOS, LIMBO and LIMES all sit in that branch.
Nothing else comes close and the drop after L is steep. So with I and M confirmed, L is the single most valuable letter to test in the first box.
TIMES, LIMIT, AIMED, TIMER and MIMIC are the entries in general use. JIMMY, DIMES and LIMOS come up often enough to matter.
E takes the fourth box in twenty one of the fifty nine and B in eleven, with A on eight and P on seven. TIMES, LIMBS, LIMAS and PIMPS show those shapes.
The B branch is the distinctive one. LIMBA, LIMBI, LIMBO, LIMBS, LIMBY, BIMBO, KIMBO and NIMBI all sit there, which is more clustering than most fourth boxes produce.
LIMIT repeats the I, which costs part of the guess. TIMES tests the E and S in the two busiest positions and is the stronger probe.
The genus that includes the bed bug. It is a scientific name rather than a word anyone would guess.
A limey is a British person, informally, while a limen is a threshold in psychology. Two unrelated words sharing four letters.
Seventeen of the fifty nine do. Most are plurals of short nouns like time, dime, limb and limo.