#Window plant shelf design

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dusty breach
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I have this piece of 7" wide, 52" long, 23/32" thick plywood leftover from a work table build that I've decided to turn into a window plant shelf. I cut 2x2's for legs/cross bits with dado/lap joints. Is this structurally sufficient, or will the plywood sag? Can I reinforce it with a 2x2 running along the bottom of the plywood, like a spine?

fallen frigate