I found the speed difference negligible, but slightly faster sprint (or brain playing tricks on me) when MIDA was equipped versus any other weapon. the speed boost is an inherent perk (was before it was made one, too!) and is active regardless.
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Wait, to clear this up, the MIDA Multi Tool agility bonus is functional on both sprint and walk, but as far as I can feel walking is faster because lightweight is broken.
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If you walk it is marginally, ever so slightly faster than sprinting. Dunemarchers and Headstrong would even it out (if not break it entirely) once both are available.
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They may fix lightweight soon anyway
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And you never know, they may fix lightweight so its the same for both walking and sprinting