  We obtain new progresses about the diameter two property and the Daugavet
property in tensor product spaces. Namely, the main results of the paper are:
  -If $X^*$ has the WODP, then $X\widehat{\otimes}_\varepsilon Y$ has the DD2P
for any Banach space $Y$.
  -If $X$ has the WODP, then $X\widehat{\otimes}_\pi Y$ has the DD2P for any
Banach space $Y$.
  -If $X^*$ and $Y^*$ have the WODP then $X\widehat{\otimes}_\varepsilon Y$ has
the Daugavet property.
  The above improve many results in the literature and establish progresses on
some open questions.
