Speaking while putting on Rabbeinu Tam's Tefillin

Is it permitted to speak between putting on the hand tefillin and the head tefillin when putting on Rabbeinu Tam's tefillin without a blessing?

It is forbidden to speak, and one who speaks commits a transgression.

Talmud, Menachot 36a: "It was taught: If one spoke between the tefillin, it is a transgression, and he returns from the ranks of war." Similarly, the Shulchan Aruch, Orach Chaim, Siman 25, Paragraph 9 rules: "It is forbidden to interrupt with speech between the hand tefillin and the head tefillin, and if one interrupted, he blesses on the head tefillin: 'on the mitzvah of tefillin'. Rema: For us who are accustomed to bless two blessings, even if he did not interrupt, he must bless again on the head tefillin 'to put on' and 'on the mitzvah' (DA)." Mishnah Berurah, Siman 25, s"k 28 states: "Even speech for the sake of a mitzvah, such as responding to a greeting from his teacher and similar cases, as it causes an unnecessary blessing. And even for those who put on tefillin during Chol HaMoed without a blessing or put on Rabbeinu Tam's tefillin after removing Rashi's tefillin or removed tefillin with the intention to return them, the Rema in s"i 12 ruled that there is no need to bless again when putting them on later, still it is a transgression to speak between them, as initially, they should be close and continuous, as it is written: 'and it shall be for a sign upon your hand and for a reminder between your eyes', that it should be one action for both."