Fretboard separating in small area.

On my inexpensive acoustic this small area of fretboard is separating from the binding.

I was thinking about thinning some titebond and masks off the area and injecting the glue into the area with a small syringe and clamping over night. Would this be an effective repair?

On my inexpensive acoustic this small area of fretboard is separating from the binding.

I was thinking about thinning some titebond and masks off the area and injecting the glue into the area with a small syringe and clamping over night. Would this be an effective repair?