Assume that I have an affine hypersurface $X =V(f)\subset \mathbb{C}^4$ of degree $d$ with an *isolated* singularity of multiplicity $m$ at the origin $o=(0,0,0,0)$. Let $$f:=f_m + f_{m+1}+ \cdots +f_d$$
be the decomposition of $f$ into homogeneous pieces and suppose, in addition, that the singularity at $o$ is *ordinary*, that is the tangent cone $C=V(f_m) \subset \mathbb{C}^4$ is a cone over a *smooth* surface of degree $m$ in $\mathbb{P}^3$.

Simple examples show that, in general, the germ $(X, o)$ is *not* isomorphic to the germ $(C, o)$. For instance, take
$$f=x_0^4+4x_0^3x_1+6x_0^2x_1^2+4x_0x_1^3+x_2^4+4x_2^3x_3+6x_2^2x_3^2+4x_2x_3^3+x_0^5+x_0x_1^4$$
$$+x_0^4x_2+7x_1^4x_2+3x_0x_2^4-x_2^5+4x_0x_3^4+11x_2x_3^4,$$
so that $$f_4=x_0^4+4x_0^3x_1+6x_0^2x_1^2+4x_0x_1^3+x_2^4+4x_2^3x_3+6x_2^2x_3^2+4x_2x_3^3.$$
Then, denoting with $\tau$ the Tjurina number, by using Singular one finds $$\tau(f, 0)=71, \quad \tau(f_4, 0)=81,$$
so the two germs are not isomorphic.
This implies that the (analytic) local rings $\mathcal{O}^{an}_{X,0}, \quad \mathcal{O}^{an}_{C,0}$
are not isomorphic, see for instance [De Jong-Pfister, Local analytic geometry, page 120].

Now my question is:

Are at least the completed rings $\widehat{\mathcal{O}}^{an}_{X,0}, \quad \widehat{\mathcal{O}}^{an}_{C,0}$ isomorphic? If the answer is yes, what is a reference? If the answer is no, what is a counterexample?

My apologies in advance to the experts of singularity theory if this question turns out to be a trivial one.

**EDIT.** Jason Starr's comments suggest that the two completed rings should *not* be isomorphic. Let me then ask a weaker form of my question:

(1)Assume that the local ring of the tangent cone at the vertex $\mathcal{O}^{an}_{C,0}$ is a UFD. Can one conclude that the same is true for $\mathcal{O}^{an}_{X,0}$?

(2)Assume that the completed local ring of the tangent cone at the vertex $\widehat{\mathcal{O}}^{an}_{C,0}$ is a UFD. Can one conclude that the same is true for $\widehat{\mathcal{O}}^{an}_{X,0}$?