The phrase "asserted though that" is not correct as it stands and may cause confusion in written English.
It can be used when you want to indicate that someone has made a claim or statement, but the word "though" is misplaced and disrupts the flow.
Example: "He asserted, though, that the results were conclusive."
Alternatives: "claimed that" or "stated, however, that".
For instance, Orth repeatedly asserts, though never quite explains, a conviction that celebrity is ruining America.
She asserts, "Though rape is certainly a serious problem, there's no evidence that it's considered a cultural norm.
The candidate asserted that though his opponent, Vice President Humphrey, claimed that Americans had never had it so good, he should tell that to the farmer.
Even that could prove difficult, though, and Democrats assert that Republicans will resort to a government shutdown to get their way.
Dr. Kissinger at a recent press conference asserted that, though this commitment was not legally binding, it was morally binding.
Discussing the possibility of stigmatization of the indigents, a health worker asserted that though the term indigent might function as a "label" within the village it did not impede the integration of individuals within the community, beyond their existing state of isolation.
Prosecutors asserted that Mr. Harris ordered that killing, though they never formally charged him with it.
Some participants in the session asserted, however, that though the Defense Department leadership may have known in advance of the proposed order, they did not.
General McNeill also asserted in an interview here, though, that Afghanistan seemed to be gaining stability faster than had been widely expected, and he suggested that a partial withdrawal of American forces was possible starting next summer if that trend continued.
Mr. Bloomberg, though, asserted that the attack had dimmed the hopes of the project's being built anytime soon.
