Let $G/K$ be a symmetric space. Let 
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, 
with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant 
differential operators in this case is commutative, and "polynomial conjecture"
states that it is isomorphic to $S(\mathfrak{p})^{\mathfrak{k}}$. 
It was formulated by C.Torossian in a 1993 paper, but actually it is a special 
case of an older conjecture by Duflo (though the reference I know are proceedings of a 1986 conference, and I haven't seen them.) 


Is this conjecture still open? If it is, it makes me a little curious, because there aren't many symmetric spaces. What are the known and the open cases then?


EDIT: The conjecture (in this form) was formulated in Torossian, C., Operateurs diﬀerentiels invariants sur les espaces symetriques I. Methodes des orbites. J. Funct. Anal. 117 (1993), no. 1, 118–173. Torossian made a reference to Duﬂo, M., in Open problems in representation theory of Lie groups, Conference on Analysis on homogeneous spaces, (T. Oshima editor), August 25-30, Kataka, Japan, 1986. (As I understand it, Duflo's conjecture is much more general; admittedly, I did not read this 1986 text). A more recent account is in "Quantification pour les paires symétriques et diagrammes de Kontsevich" A. Cattaneo, C. Torossian, Annales Sci. de l'Ecole Norm. Sup. (5) 2008, 787--852, available here <http://www.math.jussieu.fr/~torossian/>



Is the Duflo polynomial conjecture open?
As far as I know, Duflo's conjecture is still open.


Let me make several remarks:


1. Duflo's conjecture actually says that the algebra of invariant differential operators on a symetric space is isomorphic to the $\mathfrak k$-invariant part of $S(\mathfrak g)/(h-\chi(h),h\in\mathfrak k)$, where $\chi$ is the character given by half the trace of the adjoint action of $\mathfrak k$ on $\mathfrak p$. this shift by a character did not appear in Cattaneo-Torossian paper and this was very surprising... there was indeed a mistake in that paper, which is corrected in Cattaneo-Rossi-Torossian: <http://arxiv.org/pdf/1105.5973.pdf>
2. Duflo's conjecture is indeed more general. It holds for general reductive homogeneous spaces: it claims that the center of the algebra of invariant differential operators is isomorphic to the Poisson center of $\big(S(\mathfrak g)/(h-\chi(h),h\in\mathfrak k)\big)^{\mathfrak k}$.
3. Rybnikov's result mentionned in Alexander Chervov's comment prove a localized version of it for Riemaniann reductive homogeneous spaces (i.e. it holds on the level of fraction fields).


