Extendability of quadratic modules over a polynomial extension of an equicharacteristic regular local ring

We prove that a quadratic $A[T]$-module $Q$ with Witt index ($Q/TQ$)$\geq d$, where $d$ is the dimension of the equicharacteristic regular local ring $A$, is extended from $A$. This improves a theorem of the second named author who showed it when $A$ is the local ring at a smooth point of an affine variety over an infinite field. To establish our result, we need to establish a Local-Global Principle (of Quillen) for the Dickson--Siegel--Eichler--Roy (DSER) elementary orthogonal transformations.