Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S\cong R[[x_1,\cdots, x_n]]/J$ (this is a Noetherian local ring which is an $R$-algebra).
  Is the completed tensor product (https://stacks.math.columbia.edu/tag/0AMU) $S \widehat{\otimes}_R K$ a Noetherian local ring containing $K$ ?

Thoughts: Let $\mathfrak m$ be the unique maximal ideal of $S$. Since $K$ is a field, so the completed tensor product $S \widehat{\otimes}_R K$ should just be the inverse limit of the system $\dfrac{S\otimes_R K}{\mathfrak m^n \otimes_R K}$. I have no idea how to analyze this.

Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S\cong R[[x_1,\cdots, x_n]]/J$ (this is a Noetherian local ring which is an $R$-algebra) for some ideal $J$. Assume depth$(S)>0$. Is the completed tensor product (https://stacks.math.columbia.edu/tag/0AMU) $S \widehat{\otimes}_R K$ a Noetherian local ring containing $K$ ?

Thoughts: Let $\mathfrak m$ be the unique maximal ideal of $S$. Since $K$ is a field, so the completed tensor product $S \widehat{\otimes}_R K$ should just be the inverse limit of the system $\dfrac{S\otimes_R K}{\mathfrak m^n \otimes_R K}$. I have no idea how to analyze this.

Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S$ be a Noetherian local ring which is an $R$-algebra. Is the completed tensor product (https://stacks.math.columbia.edu/tag/0AMU) $S \widehat{\otimes}_R K$ a Noetherian local ring containing $K$ ?

Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S\cong R[[x_1,\cdots, x_n]]/J$ (this is a Noetherian local ring which is an $R$-algebra). 
Is the completed tensor product (https://stacks.math.columbia.edu/tag/0AMU) $S \widehat{\otimes}_R K$ a Noetherian local ring containing $K$ ?

Thoughts: Let $\mathfrak m$ be the unique maximal ideal of $S$. Since $K$ is a field, so the completed tensor product $S \widehat{\otimes}_R K$ should just be the inverse limit of the system $\dfrac{S\otimes_R K}{\mathfrak m^n \otimes_R K}$. I have no idea how to analyze this.