I wouldn't trust just tacking the thin little backer board to the cleat. You could run a couple pieces of 2×4 or something like that under that top shelf and screw the left and right sides into the 2by so that they are hanging and holding by the thin back sandwiched between the 2by and cleat
I've seen that backing fail from general use I wouldn't want to hang it just by that. Usually very small and thin nails are used to tack it on too so it would likely pop off in no time and break a large amount of your tools
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