$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What are the (lower-dimensional, say $d \leq 14$) Fivebrane bordism groups $$ \Omega_d^{\Fivebrane}=\text? $$

Given that we know the Whitehead tower

$$\require{AMScd}\begin{CD} \vdots \\ @VVV \\ B\Fivebrane \\ @VVV \\ B\String @>\frac1 6 p_2>> B^7U(1) @>\cong>> B^8\mathbb Z \\ @VVV \\ B{\Spin} @>\frac1 2 p_1>> B^3U(1) @>\cong>> B^4\mathbb Z \\ @VVV \\ B{\SO} @>w_2>> B^2\mathbb Z_2 \\ @VVV \\ BO @>w_1>> B\mathbb Z_2 \\ @VV\cong V \\ B{\GL} \end{CD}$$

and the string bordism groups:

\begin{align*} & \Omega_0^{\String}=Z \\ & \Omega_1^{\String}=Z_2 \\ & \Omega_2^{\String}=Z_2 \\ & \Omega_3^{\String}=Z_{24} \\ & \Omega_4^{\String}=0 \\ & \Omega_5^{\String}=0 \\ & \Omega_6^{\String}=Z_2 \\ & \Omega_7^{\String}=0 \\ \end{align*}

Framed bordism: $$\Omega_7^{\Fr}=Z_{240}.$$

Note that $$ \Omega_d^{\String}=\Omega_d^{\Fr}, d \leq 6. $$

Is it true that $$ \Omega_d^{\Fivebrane}=\Omega_d^{\Fr}, d \leq 7? $$ How about $$ \Omega_d^{\Fivebrane}=\text?, \text{ for } d > 7? $$