This issue is to track the work for adding the FilledInput component to @mui/material-next as part of the Material You effort (#29345)

Material You TextField (filled variant) specs: https://m3.material.io/components/text-fields/specs

Migration Guide steps

• Copy component files from material (v5) to material-next (PR)
• Remove deprecated components and componentsProps props (PR)
• Drop support for ThemeProvider in favor of CssVarsProvider (PR)
• Migrate component to Typescript (PR)
• Refactor component to use Base UI’s hook if it exists done in [Input][material-next] Add InputBase component #38392
• Implement Material You design specs (39307 but will be polished when working on TextField)
• Add playground to experiments folder (PR)
• Add component playground to v5 docs will be done with TextField
• Refactor styles to use component CSS Variables (39307 but will be polished when working on TextField)