In the title $\Phi(\cdot)$ is the usual cummulative density function of a standard normal and $a<b$ are real numbers. It is not hard to show that the function inside the logarithm reaches its maximum at $x=(a+b)/2$ and then declines fastly to $0$. Taking the $\log$ of it and plotting it we see that the function has the shape of a parabola (which makes sense intuitively).
Observe for instance the case for $a = -3$ and $b = 5$ below.
How could we go on to prove/disprove the initial claim? Thanks in advance.
EDIT: I checked it for $a,b>0$ when $x \in [(b+a)/2,b]$, now I need to verify it for the rest of the interval, using one of the post's answers.
If you already have proven it in the case $a,b > 0$, you can just make a simple substitution, i.e., $$ X = x + (-a + 1).$$ Then, $a$ is replaced by $1$ and $b$ is replaced by $b - a + 1$.