Upper bound for the complex Beta function

Question

The question is almost the same as here.

What is the upper bound for a complex Beta function $$\DeclareMathOperator{\Im}{Im}\DeclareMathOperator{\Re}{Re} \displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)} $$ where $0<\Re(s)<1$, $0<\Re(z)<1$, $\Im(s)>10$ and $\Im(z)>10$ ?

Answer 1

The command of Mathematica

NMaximize[{ComplexExpand[Abs[Gamma[s + I*t]*Gamma[a + I*b]/Gamma[s + a + I*(t + b)]]],s > 0 && a > 0}, {s, t, a, b}] 

outputs "NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded." and

{4.7857*10^8, {s -> 0.32798, t -> 0.720557, a -> 0., b -> -2.08956*10^-9}}

The command of Mathematica

s = 1/2; t = a; MaxLimit[ ComplexExpand[Abs[Gamma[s + I*t]*Gamma[a + I*b]/Gamma[s + a + I*(t + b)]]], {t,b} -> {0, 0},Direction -> "FromAbove"] 

results in $\infty$, confirming it.

Addition. Here is the answer with Mathematica to the modified question (Thanks to Brendan McKay who paid my attention to the edit.)

NMaximize[{ComplexExpand[ Abs[Gamma[s + I*t]*Gamma[a + I*b]/ Gamma[s + a + I*(t + b)]]], s > 0 && a > 0 && s < 1 && a < 1 && b > 10 && t > 10}, {s, t, a, b}] 

{1.121, {s -> 3.70155*10^-7, t -> 10., a -> 3.57795*10^-7, b -> 10.}}