Early, three-sheet plywood was often used in furniture making during the late 19th and early 20th centuries. Unfortunately, the glues used in the manufacturing process have not stood the test of time. I get a few of these types of repairs in my shop every year, and it's best to just replace the failing parts rather than fill/repair. I would typically disassemble the chair in its entirety, however the client had just had the chair reupholstered and it was too risky to disassemble - I wish she had contacted me first! 😆 I milled some mahogany to 3/8" to replicate the original plywood scroll work.