See A178160 for definition of "type-2 Trott-like constant."
Each time a type-2 Trott-like constant is extended by one digit, the size of the interval encompassing all the values that can be reached using the terms of an infinitely-extended sequence of positive one-digit numbers directly as digits (e.g., [0.375511111...,0.375599999...]) decreases by a factor of 10.
However, the size of the interval encompassing all the values that can be reached using the terms of such a sequence as terms of a continued fraction of the type used in the second Trott constant (A091694) decreases more slowly.
As a result, as the number of digits increases, the continued-fraction interval increasingly dwarfs the other interval, as is seen in comparing the intervals for a(3)=15751525912:
0+1/(5+7/(5+1/(5+2/(5+9/(1+2/(1+1/(9+9/(1+1/(9+9/...))))))))) = 0.157515259128210971946776833132...
0+1/(5+7/(5+1/(5+2/(5+9/(1+2/(9+9/(1+1/(9+9/(1+1/...))))))))) = 0.157537568512450889006911743529...
which defines an interval whose width is about 0.0000223, while the width of the interval [0.15751525912111...,0.15751525912999...] is only 0.00000000000888....
Inextensible type-2 Trott-like constants can occur only when one of the endpoints of the continued-fraction interval happens to fall inside the much smaller interval; this becomes less and less likely as the number of digits increases.
Also, because inextensible cases arise only when increasing the value of a proposed next digit would cause the continued fraction value to increase, inextensible cases must satisfy (k+1) mod 4 < 2, where k is the number of digits to the right of the decimal point. An exhaustive search found no 12-digit inextensible cases, so a(4) > 10^14.