I do not know what you want to say with "even with just an HDRi"... does an HDRi always only cast sharp shadows? As I commented, it is an optical illusion because of the very soft shadows - and the fact that the colors have a similar hue. A darker brown slat with a soft shadow fading into a lighter brown background looks like a gradient from dark to light, and this is creating an optical illusion as if it was blurry - because a blurred image would soften borders to a kind of gradient between those colors. But it is not really blurry.
So here is your image with the very soft shadows:
And now just a very simple change, reducing the ridiculously large radius of the point lights from 50 m to 1 m (which is still large but your scene is scaled very large as well). This will make the shadows a bit sharper, especially on the slats further away from the center and reduce the illusion of blurriness a bit:
Of course this also makes the light less evenly spread. But just to make it totally clear that there is nothing blurry, I gave the wall a white emissive material so it cannot receive any shadows, and I made the slats completely black. There is no blurriness apart from some antialiasing:
And even with your totally large lights throwing soft shadows you can make it look less blurry if the slats would be contrasting the wall more, not only regarding the brightness, but for example if they had a completely different hue so that the shadows would not look like as if the slats are blurred with the wall. Like in this example, the blue is not blurred with the light brown background: